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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 7
Chapter 4, Problem 7

Divide using long division. State the quotient, and the remainder, r(x)r(x). (12x2+x−4)÷(3x−2)(12x^2+x-4)\(\div\)(3x-2)

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Identify the dividend and divisor: the dividend is \(12x^{2} + x - 4\) and the divisor is \(3x - 2\).
Set up the long division by writing \(12x^{2} + x - 4\) under the division bar and \(3x - 2\) outside.
Divide the leading term of the dividend, \$12x^{2}\(, by the leading term of the divisor, \)3x$, to find the first term of the quotient: \(\frac{12x^{2}}{3x} = 4x\).
Multiply the entire divisor \(3x - 2\) by \$4x$ and subtract the result from the dividend to find the new remainder.
Repeat the process: divide the leading term of the new remainder by \$3x$, multiply the divisor by this term, subtract again, and continue until the degree of the remainder is less than the degree of the divisor.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by this result, subtracting, and repeating until the degree of the remainder is less than the divisor.
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Introduction to Polynomials

Quotient and Remainder in Polynomial Division

When dividing polynomials, the quotient is the polynomial result of the division, and the remainder is the leftover polynomial with a degree less than the divisor. The division can be expressed as Dividend = Divisor × Quotient + Remainder.
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Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. Understanding degrees is essential in polynomial division because the process continues until the remainder's degree is less than the divisor's degree.
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Related Practice
Textbook Question

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.

f(x)=x2+2x+1g(x)=x2−2x+1f(x) = x^2 + 2x + 1 \(\quad\) g(x) = x^2 - 2x + 1

h(x)=x2−1j(x)=−x2−1h(x) = x^2 - 1 \(\quad\) j(x) = -x^2 - 1

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Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x2+5x+4>0

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Textbook Question

In Exercises 5–6, use the function's equation, and not its graph, to find (a) the minimum or maximum value and where it occurs. (b) the function's domain and its range. f(x)=−x2+14x−106f(x) = -x^2 + 14x - 106

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Textbook Question

In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x4−11x3−3x2−6x+8

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Textbook Question

Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies jointly as x and z. y = 25 when x = 2 and z = 5. Find y when x = 8 and z = 12.

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Textbook Question

Determine which functions are polynomial functions. For those that are, identify the degree. f(x)=x12−3x2+5f(x)=x^{\(\frac\)12}-3x^2+5

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